MATHEMATICS · 2019. 2. 25. · Edited by Professor Oliver Gottschalg PRIVATE EQUITY MATHEMATICS Cover.PEM 20/3/09 12:30 Page 1

Edited by Professor Oliver Gottschalg

PRIVATE EQUITYMATHEMATICS

Cover.PEM 20/3/09 12:30 Page 1

IntroductionIt is generally agreed that portfolios go through the classic life cycle of construct,nurture and harvest,1 a process about which there is abundant research and viewson managing various types of investment portfolios. Most of the research expoundsupon the general principals articulated by Harry Markowitz in the Journal of Finance(1952) that serves as the foundation of the Modern Portfolio Theory (MPT). Thetechnical research assumes that a portfolio is comprised of assets that are amongother things fungible, transparent, readily quoted and easily transferable. These ele-ments contribute towards understanding the risk-reward trade-offs between theinvestment choices thereby allowing the portfolio manager to craft an appropriateportfolio given their individual utility function.

The private equity (PE) market raises very unique challenges, including:

• Construct phase – lack of unitised/clean data, non-uniform access with generallylarge minimums, cash flow uncertainty and multi-year commitments, qualitativeaspects (e.g., talent, relationships etc.) and other such elements.

• Nurture phase – lack of ability to actively manage or influence could vary frombeing completely passive for limited partners (LP) to being active for general part-ners (GP). However, in post-portfolio construction (or target acquisition) even themost active GPs can do little other than continue to be active on the individualportfolio companies themselves.

• Harvest phase – lack of multiple or defined exit options imply realisations could besub-optimal or span many years. Continuing development of the secondary mar-kets, structured products and listed PE funds notwithstanding, exit options arequite limited thereby making the asset class illiquid.

Further, the PE industry as a whole is not known to maintain very robust datasets,which includes issues such as lack of depth, information lag, lack of true price dis-covery, as well as selection and self-reporting biases. Reported returns are not nor-mally distributed and also capital weighted thereby making uniform, unitisedallocation analysis very difficult. It can also be generally agreed that possibly themost important aspect of PE portfolio management is the upfront selection whetherit be the investment with a GP or the investment that a GP makes. Therefore, giventhe uniqueness of the PE markets, data issues and overlay of multiple non-quantifi-

Private equity as part of your portfolioBy Satyan Malhotra, Caspian Capital Management

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Chapter 8_Malhotra.PEM 20/3/09 11:48 Page 81

able elements, PE portfolio management is as much an art as a science. However,given various non-technical elements, even if it is not possible to clearly articulatethe exact methods of portfolio management, it may be possible to identify some gen-eral parameters, principals and metrics (herein, collectively called the ‘private equi-ty tools’ (PET)). The potential application of the PETs in managing PE portfolios isunique to the type of participant, where:

• GPs focus on industry sub-sectors2 (e.g., IT, industrial, etc.)• Fund of funds focus on various types of GPs (e.g., buyout, secondary, etc.)• Investors focus on types of investments (e.g., PE, public equity, bonds, etc.)

This chapter begins by presenting select PETs and then performs sample analysesfrom the perspective of each PE participant (i.e., GP, fund of funds and investors).The overall intent of the chapter is to demonstrate methods of estimating the PETsas well as highlighting PE participant-specific illustrations and presentation styles.At the onset, it is also equally important to remind the reader of the numerous con-cerns highlighted previously, therefore the results should be used with extreme cau-tion and more so as relative anchor points used with some degrees of freedom.

Private equity tools (PETs)As with all market practitioners, PE participants have their own preferences aboutthe tools they use for portfolio management. However, although the tools, exact for-mulae and their utility may vary across the practitioners the analysis itself can begrouped into three general categories: (a) return-related, (b) risk-related, and (c)portfolio level. This section presents select PETs and their estimation formulae foreach of the three general categories.

Return-relatedExpected return is the mathematical expectation of return from a single or portfolioof holdings generally based on the expected probability of each return. This isexpressed as follows:

whereRi is the possible return and Pi is the possibility of that outcome.

In quantifying the expected return, it is also important to establish the parametersaround the expected return or whether it is: (a) relative or absolute, and (b) cash-on-cash or in percentages e.g., a 2x multiple return is 41 percent IRR if cash isreturned in year two versus 10 percent IRR if cash is returned in year seven.

E(R) = ∑PiRii=1

n

Investing in private equity

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Mean return is the arithmetic average of the return. Weighted average mean returnwould include an additional set of information along with the return for the holdinge.g., assets, number, capital, etc.

Quartile is the measure of the relative ranking of the holding (e.g., return). The Kthquartile of population X can be defined as the value ‘x’ such that:

Risk-relatedStandard deviation is the measure of deviation of values from their mean. It can beused as a measure of risk appetite or the tolerance of volatility on a single and/or aportfolio of investments. The more recent accounting pronouncements (e.g., USGAAP FAS 157) to mark holdings to market would create further intra-period valua-tion fluctuations.

Variance is the measure of dispersion around a value (e.g., expected return).

Semi-deviation is the measure of volatility of values that occur below a target return.

whereT is the target return, and for this chapter the target return is assumed to be 8percent.

[max (0, T - xi)]2

i=1

n

n∑

Var(x)= ∑(xi -i=1

n

x)2

n

σx= ∑(xi -i=1

n

x)2

n

P(X≤x)≤p and P(X≥x)≥1 - p

where p= k4

x= ∑wixii=1

n

where

∑wi = 1i=1

n

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Private equity as part of your portfolio


Loss probability is the measure of the probability of loss in the portfolio. Some lossprobability (SLP) is the ratio of results whose return is negative to the total numberof returns. Assuming M possible results from a simulation, and S of them have neg-ative returns, then the SLP is 100 (S/M) percent. Similarly, total loss probability(TLP) is the ratio of numbers of negative one return to the total number.

Portfolio levelBarbell strategy is portfolio composition with extreme holdings. For example, assum-ing an allocation to two positions then one position each in lower risk-return andhigher risk-return assets (e.g., equal allocation to 1.5x and 2.5x assets with expect-ed value at 2x).

Bullet strategy or concentrated strategy refers to portfolio composition withmid-valueor target holdings. For example, assuming an allocation to two positions then bothpositions in mid risk-return assets (e.g., equal allocation to 2x and 2x assets withexpected value at 2x). Further, tolerance thresholds can drive the level of skewnessin the portfolio.

Correlation is the measure of relationship between two values. Estimates range from-1 (exact opposite relationship) to +1 (exact relationship).

wherex , y are the mean of xi, yi , and σx, σy are the standard deviation of xi, yi.

Intra-portfolio correlation is the measure of diversification within a portfolio or thedegree of correlation of holdings within the portfolio.

whereXi, Xj are the fraction invested in asset i, j, and Pij is the correlation between asseti and j.

Skewness is the measure of symmetry of the holdings within the portfolio. Positivemeans elongated right tail and negative means elongated left tail.

r1 =∑(xi -i=1

n

x)3

nσx3

Q=∑∑XiXjPiji=1

n

j=1

n

∑∑XiXji=1

n

j=1

n

ρxy=∑xiyi -nx y(n - 1)σxσy


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Kurtosis is the measure of the peakedness of the holdings within the portfolio. Highkurtosis implies distinct peak with rapid decline. Low kurtosis implies flat top withslow decline.

Random variable is the sequence of outcomes that do not have a defined pattern.There are many methods of generating random variables and some have been auto-mated by third-party vendors like Matlab 3 (financial applications software from TheMathWorks). One is the method suggested by D.H. Lehmer, where the generatorshave three integer parameters: a, c and m, and an initial value x0, called the seed. Asequence of integers is defined by:

The operation ‘mode m’ means taking the reminder after division bym.

Sharpe ratio is the measure of the derived risk premium per unit of risk. HigherSharpe ratios imply greater relative risk-adjusted value.

whereRf is the risk-free rate. For this chapter the risk-free rate is assumed to be 3 percent.

Sortino ratio is the measure of the risk-adjusted return of the portfolio. As a modifi-cation of the Sharpe ratio, it measures the derived risk premium over a target return.

whereT is the target rate. For this chapter the target return is assumed to be 8 percent.

The subsequent sections estimate these PETs using defined sources of datasets foreach of the PE participants. The dataset can be analysed on a standalone or on a sim-ulated basis. Standalone analysis would mean that the dataset itself is used for esti-mating these PETs. Simulation would mean that the dataset is used as a referencepoint by a random process that consequently builds a distribution of results that isused for estimating the PETs. For this chapter, the Matlab software has been used forsimulating the expected-return distributions from the sample datasets over

r4 =E(r) - T

Semi-deviation

r3 =E(R - Rf)

Var(R - Rf)

xk+1 = axk+ c mod m

r2 =∑(xi -i=1

n

x)4

nσx4

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[200,000] iterations. Thereafter the expected-return distributions were used to cal-culate the PETs based on the aforementioned formulae. For simplicity purposes, itwas assumed that the datasets are perfectly unitised with the ability to make equalallocations on each iteration.

General partner perspectiveThe previous section identified select PETs that can be used for portfolio managementwhile this section examines their application from a GP’s perspective. GPs use a top-down (driven by macroeconomic factors or themes), bottom-up (opportunistic devel-opment) or a combination approach inmanaging their PE portfolios. Overall, portfoliomanagement techniques will anchor around the composition of the GP’s talent pool,which includes: (a) investment access, (b) ability to add value, (c) amount of dispos-able capital, (d) breadth and depth of team, and (e) incremental capital need/access.For example, GPs managing larger funds may devote resources to develop sector spe-cific teams/themes but, given their fund size they generally do not participate in small-er deals. On the other hand, GPs managing smaller funds can remain specialists butthey may trade-off on the size of the deal or on control. So, portfolios mean differentthings to different GPs, where there can be various quantifiable as well as non-quan-tifiable reasons for allocating capital across and within the different companies or sec-tors in the portfolio.

The PETs can help GPs perform quantitative sector analysis. In conducting this analy-sis, given the previously articulated data issues, the private market dataset was sup-plemented with the public market sector data as a proxy. For this section, the twodatasets have (a) only included sectors where both datasets hadmore robust informa-tion, and (b) been organised into the following four sector groups to be better aligned.

• Industrial includes energy-related, chemicals, construction materials, metals,packaging materials, capital goods, including aerospace and defence, construc-tion, engineering & building products and industrial machinery.

• Consumer includes food, beverages, personal products, supermarkets, retail,apparel, hotels & restaurants, media production and services, durable and non-durable household goods.

• Healthcare includes providers of healthcare products, services & operators ofhealthcare facilities, R&D,marketing & production of pharmaceuticals and biotech.

• IT includes software, hardware, database management, IT consulting & services,Internet, telecom (fixed and wireless), semiconductors.

It is possible to use different datasets and time periods to organise this data into small-er or larger groups as well as to include other sectors or groupings. For these illustra-tions, the information included the one-year, three-year, five-year, ten-year, 15-yearand 20-year sector returns that were sourced private and public market returns.


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• Private market returns of US pooled industry net IRRs from Thomson Financial(beginning in 1991) which include the information housed by Thomson Financialon all private equity transactions (e.g., buyouts, mezzanine, and venture capital)as sourced in July 2008.

• Public market returns of US industry indices from MSCI (beginning in 1996). Thethree-year, five-year, and ten-year cumulative industry returns provided by MSCIwere annualised.

Return-relatedThis generally varies across sectors, time periods or horizons and economic cycles. It isnot unusual for certain sectors to have significant relative over-performance over cer-tain periods of time (e.g., IT in 1998 or 1999). Figures 8.1 and 8.2 show the one-year,three-year, five-year and ten-year historical returns across the four pre-defined sectors.


Figure 8.1: Historical returns across pre-defined sectors – private market

Return

(%)

-5

0

10

30

5

20

25

15

3 year(1997–2007)

5 year(1997–2007)

10 year(1997–2007)

1 year(1997–2007)

ITHealthcareConsumerIndustrial

Figure 8.2: Historical returns across pre-defined sectors – public market

Return

(%)

-5

0

10

30

5

20

25

15

3 year(1998–2007)

5 year(2000–2007)

10 year(2005–2007)

1 year(1997–2007)

ITHealthcareConsumerIndustrial

87


Figure 8.3: Quartile returns, 1 year (1997–2007) – privateRe

turn

(%)

-50

-30

10

110

-10

50

70

90

30

Consumer Healthcare ITIndustrial

Mean100%75%50%25%0%

Figure 8.4: Quartile returns, 1 year (1997–2007) – public

Return

(%)

-50

-30

10

110

-10

50

70

90

30

Consumer Healthcare ITIndustrial

Mean100%75%50%25%0%

Figure 8.5: Expected return distribution, 1 year (1997–2007) – private

Simulation20

0,00

0

0

2,000

6,000

16,000

4,000

10,000

12,000

14,000

8,000

-50 -30 -10 10Percent

30 50 70 90-70

4 sectors1 sector


88


Figures 8.3 and 8.4 show that there is significant dispersion in the returns and that top-quartile returns exhibit high over-performance when compared to the mean returns.

Figure 8.5 shows that the expected return distribution takes on a more familiar ordefined profile as more sectors are included in the portfolio. The expected returndistributions can be constructed using: (a) random sector selection, or (b) anchor-ing the first sector and then introducing one more sector at a time. The results of thesecond approach should gradually approach the results of the first approach as moresectors are included. The deviation within the approaches will generally depend onthe correlation between the sectors.

Risk-relatedStandard deviation generally reduces as more sectors are included in the portfolio.Figures 8.6 and 8.7 show that this is true for both private and public portfolios.

Figure 8.6: Standard deviation – privatePe

rcen

t

0

5

15

35

10

25

30

20

2 sectors 3 sectors 4 sectors1 sector

10 year (1997–2007)5 year (1997–2007)3 year (1997–2007)1 year (1997–2007)

Figure 8.7: Standard deviation – public

Percen

t

0

5

15

35

10

25

30

20

2 sectors 3 sectors 4 sectors1 sector



89


However, diversification benefits are not limitless and it is important to be cognizantof the marginal impact of standard deviation versus return so as not to over diversi-fy and return to mean.

Portfolio levelCorrelation varies across sectors, time periods or horizons as well as vintages. Table8.1 shows the correlation between the one-year returns of the PE sectors and Table8.2 shows the correlation between the returns of the various holding periods for theindustrial sector.

Since public and private datasets have been used, Table 8.3 shows the correlationbetween the sectorswithin eachgroup.Table8.4measures the intra-portfolio correlationand shows that the individual sectors provide some level of diversification to each other.

Other metrics also vary across sectors, time periods or horizon as well as vintages.Tables 8.5 and 8.6 show that both data-series have a positive skew and some Kurtosis.Further, the Sharpe and Sortino ratios become better as more sectors are added.

Fund of fund portfolio perspectiveThe previous sections examined select PETs, their estimation formulae as well astheir application for GPs. This section looks at PET application for fund of funds orfor that matter investors looking at developing PE programmes (e.g., PE allocations

Table 8.1: Correlation between the one-year returns of the PE sectors

Industrial Consumer Healthcare IT

Industrial 1.00 0.57 0.48 0.53

Consumer 0.57 1.00 0.92 0.94

Healthcare 0.48 0.92 1.00 0.96

IT 0.53 0.94 0.96 1.00

Table 8.2: Correlation between the returns of the various holding periods for theindustrial sector

1 year 3 year 5 year 10 year

1 year 1.00 0.78 0.71 0.48

3 year 0.78 1.00 0.88 0.70

5 year 0.71 0.88 1.00 0.80

10 year 0.48 0.70 0.80 1.00


90


across many GPs). This section will estimate the PETs across the three previouslyarticulated categories (i.e., return-related, risk-related and portfolio level).

Table 8.3: Correlation between sectors within each group

Industrial –public

Consumer (a) –public

Healthcare –public IT – public

Industrial – private 0.77 0.21 -0.01 0.19

Consumer (a) – private 0.71 0.82 0.78 0.85

Healthcare – private 0.40 0.54 0.65 0.59

IT – private 0.55 0.72 0.72 0.77

Table 8.4: Intra-portfolio correlation – some sectors provide some level ofdiversification to each other

1 year 3 year 5 year 10 year

1 sector 1.00 1.00 1.00 1.00

2 sectors 0.87 0.78 0.74 0.75

3 sectors 0.82 0.71 0.66 0.67

4 sectors 0.80 0.68 0.61 0.62

Table 8.5: Data-series have a positive skew and some Kurtosis – private

No of sectors Semi-dev (%) Skewness Kurtosis Sharpe Sortino

1 12.52 1.44 6.15 0.58 1.35

2 7.76 1.02 4.55 0.81 2.19

3 5.81 0.83 4.03 0.99 2.91

4 4.66 0.72 3.77 1.15 3.63

Table 8.6: Data-series have a positive skew and some Kurtosis – public

No of sectors Semi-dev (%) Skewness Kurtosis Sharpe Sortino

1 12.39 0.20 3.61 0.40 0.19

2 8.48 0.15 3.29 0.57 0.28

3 6.77 0.11 3.21 0.70 0.34

4 5.75 0.08 3.14 0.81 0.41


91


In estimating the PETs, instead of looking at the profiles of individual fund of funds,this section uses underlying GP return information as a proxy for constructing/eval-uating fund of funds. For this illustration, the dataset included the individual finalfund net IRRs from 1991 through 2007 vintages. The dataset was sourced fromPreqin in July 2008 and organised into the following fund categories:

• Buyout: 433 data points for US and 613 data points for global• Secondaries: 39 data points for US and 46 data points for global• Expansion/venture: 599 data points for US and 820 data points for global• All components: 1071 data points for US and 1479 data points for global

It is possible to use different datasets and time-periods to organise this data intosmaller or larger categories as well as to include other categories.

Figure 8.8: Average quartile IRR depending on number of funds – US buyout

Net

IRR(%

)

-150-100

50

350

-50

100150200250300

0

5 10No of funds

15 201

Linear (0%) Linear (100%)100%75%50%25%0%

Figure 8.9: Average quartile IRR depending on number of funds – global buyout

Net

IRR(%

)

-150-100

50

350

-50

100150200250300

0

5 10No of funds

15 201

Linear (0%) Linear (100%)100%75%50%25%0%


92


Return-relatedReturns have a high degree of dispersion and vary across vintages, categories andnumber of fund allocations. Figures 8.8 and 8.9 show that the dispersion of returnsis greatly reduced as the number of funds is increased (as long as the funds do nothave a high degree of positive correlation). Figures 8.10 and 8.11 show that the dis-persion within the categories is even more pronounced with expansion/ventureexhibiting the highest level of dispersion. Note that Figure 8.11 illustrates the peakand bottom bias highlighted as a shortfall of the Matlab tool, where the peak of theall dispersion is not bounded by the sub-categories. Irrespective, it illustrates thehigh dispersion levels.

Figures 8.12 and 8.13 show that fund vintage plays amajor role in portfolio perform-ance. Although the investment periods of funds provide a high degree of flexibility

Figure 8.10: Investment type returns – US

Net

IRR(%

)

-40-200

160

20

6080

120100

140

40

Secondaries Expansion +Venture

AllBuyout

Mean100%75%50%25%0%

Figure 8.11: Investment type returns – global

Net

IRR(%

)

-40-200

160

20

6080

120100

140

40

Secondaries Expansion +Venture

AllBuyout

Mean100%75%50%25%0%


93


to allocate capital over larger time-horizons, the same vintage funds will generallyhave some biases including macroeconomic conditions, credit conditions and oppor-tunity set. Conversely, as illustrated in three-years-plus, allocating capital across vin-tages increases the odds of capturing vintage performance biases. Figure 8.14 showsthat the expected return distribution takes on a more familiar or defined profile asmore funds are added.

Risk-relatedStandard deviation reduces as more funds are included in the portfolio. As illus-trated in Figures 8.15 and 8.16, standard deviation falls but beyond approximate-ly 20 funds the marginal impact becomes less pronounced. Given that most fundshave between ten and 20 companies within their portfolio, at a 20-fund level afund of funds is expected to be diversified across 200 to 400 companies. Optimal

Figure 8.12: Vintage returns – USNet

IRR(%

)

-40

0

40

200

80

160

120

2 year(1993–94)

3 year(1993–95)

4 year(1993–96)

5 year(1993–97)

6 year(1993–98)

7 year(1993–99)

1 year(1993)

Mean100%75%50%25%0%

Figure 8.13: Vintage returns – global

Net

IRR(%

)

-40

0

40

200

80

160

120

2 year(1993–94)

3 year(1993–95)

4 year(1993–96)

5 year(1993–97)

6 year(1993–98)

7 year(1993–99)

1 year(1993)

Mean100%75%50%25%0%


94


Figure 8.14: Expected return distribution – 2001 vintage yearSimulation100,00

0

0

2,000

6,000

16,000

4,000

10,000

12,000

14,000

8,000

-80 -60 -40 -20Percent

0 20 40 60 80 100 120 140-100

20 funds1 fund

Figure 8.15: Standard deviation – US

Percen

t

0

10

30

60

20

50

40

5 funds 10 funds 15 funds 20 funds1 fund

SecondariesBuyoutAllExpansion + Venture

Figure 8.16: Standard deviation – global

Percen

t

0

10

30

60

20

50

40


SecondariesBuyoutAllExpansion + Venture


95


management would focus on ensuring that the fund of funds composition has builtin diversification across the fund allocations so as not to over diversify or converse-ly be mislead into believing in the diversification because of sheer numbers (e.g.,total allocation to generalist that in the end have a high sector bias due to the pre-vailing opportunity set). Further, making too many fund allocations always posesthe risk of returning the expected return of the portfolio to the mean expectedreturn of the market, which would be substantially under the top-tier return.

Figures 8.17 and 8.18 show that the risk of loss falls as more funds are included. Thisis in line with the estimates provided in Figures 8.15 and 8.16 and as such also high-lights the relative higher risk profile of the GP versus a fund of funds investment.

Portfolio levelOther metrics also vary across categories, return periods as well as vintages. Table

Figure 8.17: Risk of loss – USPe

rcen

t

0

5

15

35

10

25

30

20


Some lossTotal loss

0.17%

Figure 8.18: Risk of loss – global

Percen

t

0

5

15

35

10

25

30

20


Some lossTotal loss

0.29%


96


8.7 shows that the data-series have a positive skew and some categories a more pro-nounced level of Kurtosis. Further, the Sharpe and Sortino ratios become better asmore fund and categories are added.

Investor portfolio perspectiveThis section examines PETs for investors that are evaluating diversifying across thetraditional more liquid asset classes (e.g., public equity, bonds etc.) into PE. This sec-tion estimates the PETs pre-/post-inclusion of PE within a portfolio from the threepreviously articulated categories (i.e., return-related, risk-related and portfolio

Table 8.7: Data-series have a positive skew and some categories have morepronounced level of Kurtosis

No of funds Semi-dev(%) Skewness Kurtosis Sharpe Sortino

Buyout – US

1 9.60 1.27 12.25 0.58 0.74

5 3.03 0.53 4.77 1.29 2.34

10 1.52 0.39 3.93 1.83 NM

15 0.91 0.32 3.57 2.24 NM

20 0.62 0.29 3.50 2.59 NM

Secondaries – US

1 2.33 0.08 2.03 1.48 NM

5 0.12 0.03 2.81 3.32 NM

10 0.01 0.02 2.90 4.68 NM

15 0.00 0.03 2.96 5.76 NM

20 0.00 0.01 2.94 6.62 NM

Expansion + Venture US

1 15.85 5.31 45.13 0.19 0.27

5 7.98 2.34 11.07 0.43 0.54

10 5.72 1.68 7.24 0.60 0.74

15 4.56 1.35 5.66 0.75 0.94

20 3.84 1.18 5.08 0.86 1.12

All – US

1 13.40 5.80 60.64 0.28 0.44

5 5.51 2.58 14.73 0.63 1.07

10 3.49 1.81 8.61 0.89 1.68

15 2.59 1.46 6.67 1.09 2.27

20 2.04 1.27 5.80 1.26 2.88


97


level). The basic PETs and therein their limitations afforded to the GPs and fund offunds remain the same as to the other investors.

For this analysis, the one-year, three-year, five-year, and ten-year return data wassourced from:

• Private equity returns of pooled industry net IRR from Thomson Financial whichincludes the information housed by Thomson on all private equity transactions(e.g., buyouts, mezzanine, and venture capital) as sourced in July 2008.

• MSCI World Index returns exclude the US market and is based on data fromMSCI Barra.

• Dow Jones Industrial Average returns are based on data from Dow Jones Indexes,a unit of Dow Jones & Company.


Figure 8.19: Historical return profile of selected asset classes

Return

(%)

0

10

30

5

20

25

15

3 year(1997–2007)

5 year(1997–2007)

10 year(1997–2007)

1 year(1997–2007)

MSCI LB (US) LB (Global excl. US)DJPE (EU)PE (US)

Figure 8.20: Impact on the PE return when other asset classes are added

Return

(%)

0

10

30

5

20

25

15

3 year(1997–2007)

5 year(1997–2007)

10 year(1997–2007)

1 year(1997–2007)

PE (US + EUR) + DJ + MSCI PE (US + EUR) + DJ + MSCI + LB (all)PE (US + EUR) + DJPE (US + EUR)

98


• Lehman Brothers Index returns are for US and World excluding US and are basedon data from Lehman Brothers. The three-year, five-year, and ten-year returnswere extrapolated from the one-year returns provided by Lehman Brothers.

It is possible to use different datasets, proxies and time-periods to organisethis data into smaller or larger datasets as well as to include other asset classesor indices.

Return-relatedReturns vary over time and economic cycles. Figure 8.19 shows the historical returnprofile of the selected asset classes with PE generally over performing the otherclasses. Figure 8.20 shows the impact on the PE return when other asset classes areincluded in the portfolio.

Figure 8.21: Dispersion of return

Return

(%)

-40

-20

0

100

20

60

80

40

PE (EU) DJ MSCI LB (US) LB(Global excl. US)

PE (US)

Mean100%75%50%25%0%

Figure 8.22: Impact on dispersion of return

Return

(%)

-40

-20

0

100

20

60

80

40

PE (US + EUR) + DJ PE (US + EUR)+ DJ + MSCI

PE (US + EUR) +DJ + MSCI + LB (all)

PE (US + EUR)

Mean100%75%50%25%0%


99


Figure 8.21 shows the dispersion of return, where the bond markets are more tightlydistributed followed by the public equity markets and then the private equity markets.Figure 8.22 shows the impact on dispersion of return as more asset classes are includ-ed in the portfolio.

Risk-relatedStandard deviation generally reduces as more asset classes are added, as shown inFigure 8.23. Some care must be taken in accounting for the PE standard deviationssince the pooled IRR series may not be updated very frequently.

Portfolio levelCorrelation varies between asset classes and period of returns, as shown in Table 8.8As expected, PE markets are more closely correlated with the public equity marketsthan bond markets. An additional item to evaluate further would be that the pooledIRRs used to estimate the PE market returns are generally event-based realisations,which would tend to correlate with the public markets.

Other metrics vary across asset classes and return periods. The analysis capturesthe basic portfolio management essence of comingling assets that tend to behave dif-ferently. Table 8.9 shows that when included, PE tends to have a positive impact onthe portfolio.

ConclusionWith a cautionary look towards the theoretical purists, this chapter stated upfrontthat the multitude of data issues, qualitative overlay and other non-fungible hold-ings aspects make PET estimation challenging. Data analysis from three different

Figure 8.23: Standard deviation reduces as more asset classes are addedPe

rcen

t

0

5

15

30

10

25

20

PE (US + EUR) + DJ PE (US + EUR)+ DJ + MSCI

PE (US + EUR) +DJ + MSCI + LB (all)

PE (US + EUR)

AllExpansion + VentureSecondariesBuyout



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perspectives reveals that in spite of the aforementioned shortfalls the results gener-ally tend to be in line with a PE market practitioner expectations.

• Return-related: have a high degree of dispersion and vary across holdings (e.g.,sectors, funds, vintages, etc.). Top-quartile performers significantly outperformmean returns.

• Risk-related: diversification across lesser correlated assets tends to reduce portfo-lio volatility as well as risk of loss.

• Portfolio level: PE investments generally tend to be the return enhancer and on arisk-adjusted basis inclusion of PE in the portfolio seems to imply greater value.

Further, for GPs, funds of funds and investors there are some noteworthy partici-pant-based observations from the PET analysis.

Table 8.8: Correlation varies between asset classes and period of returns

PE (US) PE (EU) DJ MSCI LB USLB (globalexcl. US)

PE (US) 1 0.8071 0.7270 0.6812 -0.6855 -0.3572

PE (EU) 0.8071 1 0.504 0.4775 -0.5568 -0.5350

DJ 0.7270 0.5040 1 0.8002 -0.4972 -0.0495

MSCI 0.6812 0.4775 0.8002 1 -0.7443 0.1963

LB US -0.6855 -0.5568 -0.4972 -0.7443 1 0.2374

LB (global excl. US) -0.3572 -0.5350 -0.0495 0.1963 0.2374 1

Table 8.9:When included, PE tends to have a positive impact on the portfolio

Semi-dev (%) Skewness Kurtosis Sharpe Sortino

PE (US) 10.52 0.60 4.19 0.66 0.91

PE (EU) 7.02 0.04 2.12 0.94 1.97

DJ 10.20 -0.20 1.77 0.34 -0.03

MSCI 13.92 -0.35 1.91 0.28 0.00

LB (US) 3.71 -0.38 2.19 0.93 -0.44

LB (global excl. US) 9.44 0.06 1.42 0.24 -0.24

PE (US + EUR) 8.95 0.32 3.23 0.79 1.31

PE (US + EUR) + DJ 9.40 0.49 3.52 0.64 0.82

PE (US + EUR) + DJ + MSCI 10.66 0.33 3.44 0.55 0.55

PE (US + EUR) + DJ + MSCI+ LB (all)

9.65 0.65 4.36 0.48 0.34


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For GPs:• Year-on-year sector performance is disparate and impacted by macro-events, cred-it conditions, market sentiment, prevailing opportunity set, etc. Diversificationacross sectors generally increases balanced risk-return possibilities.

For fund of funds:• Fund return quartiles are very disparate but dedicated programmes (e.g., in-house or third-party) with clear strategies should be able to push the portfolio totop-tier returns.

• Diversification across approximately 20 managers balances the risk-return profileand also significantly reduces the risk of loss. However, the programme shouldaccount for the company-level diversification attained by the underlying GPinvestments to ensure that the fund is not overly diversified to drive returns to thesector or PE market’s mean return.

• It may not be possible to time the PEmarket performance but generally, multi-yearallocations, diversification across investment types and backing historic alpha gen-erators all increase the odds of top-tier performance.

For investors:• Inclusion of PE with other asset classes should diversify the portfolio and increasethe potential for higher returns.

In conclusion, although a blind, highly quantitative approach to PE managementmay not be most appropriate, general application of the PETs provides some usefulinsights. Hopefully this chapter introduced readers to some base PET applications aswell as sample illustrations as a means to present the results. The illustrated PETs,datasets, analysis, assumptions should be used as building blocks by readers to cre-ate various permutations. ■■

1 Depending on trading or maturity strategies the portfolios may be with or without compositionchurn during the holding period.

2 For the purposes of this chapter the focus is on the industry sub-sector as a whole rather uniqueopportunities within the sub-sector.

3 It should be pointed out that some of the shortfalls of using the Matlab package include::

• Random number bias: In Matlab, the computing is deterministic based on the machine codeunless linked to external devices like a gamma ray counter. So the random numbers generatedduring the simulation may have deterministic sequences. Although each number is expected tobe uniformly distributed, the blocks containing 20 contingent numbers from this sequencemaybe biased.

• Peak and bottom bias: in the quartile analysis, the 0 percent and 100 percent quartile may notbe stable as the simulation cannot generate all the possible results. For example, in the fund offunds analysis, for a 20-fund portfolio randomly selected from a 1,000-fund pool, theoreticallyyields 1,00020 = 1060 possible portfolios. Given the [200,000] iteration simulation the bottomand peak may show biases in such large pool sizes.


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Satyan Malhotra is the president of Caspian Capital Management, LLC (Caspian) a New York-based invest-ment adviser that, as of December 2008, managed $822 million in hedge fund strategies; and $490 million inits fund of private equity funds activities. In addition, Caspian provides non-discretionary services for $3.4 bil-lion in fund of hedge fund assets.Mr.Malhotra is also the COOandpart of the Investment Committee of NatixisCaspian Private Equity, LLC, also a New York based investment adviser, which, as of December 2008, managed$201 million in PE Fund-of-Funds and Direct Investment portfolios. Prior to Caspian, Mr. Malhotra was a seniormanager with the Global Risk Management Solutions practice of PricewaterhouseCoopers. Mr. Malhotra is aFinancial Risk Manager – Certified by the Global Association of Risk Professionals, MBA from Virginia Tech anda BA (Hons) Economics from University of Delhi.


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MATHEMATICS · 2019. 2. 25. · Edited by Professor Oliver Gottschalg PRIVATE EQUITY MATHEMATICS Cover.PEM 20/3/09 12:30 Page 1